Advertisements
Advertisements
प्रश्न
\[\frac{1}{\sqrt{x}}\]
Advertisements
उत्तर
\[ \frac{d}{dx}\left( f(x) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[ = \lim_{h \to 0} \frac{\frac{1}{\sqrt{x + h}} - \frac{1}{\sqrt{x}}}{h}\]
\[ = \lim_{h \to 0} \frac{\sqrt{x} - \sqrt{x + h}}{h\sqrt{x}\sqrt{x + h}} \times \frac{\sqrt{x} + \sqrt{x + h}}{\sqrt{x} + \sqrt{x + h}}\]
\[ = \lim_{h \to 0} \frac{x - x - h}{h\sqrt{x}\sqrt{x + h}\left( \sqrt{x} + \sqrt{x + h} \right)}\]
\[ = \lim_{h \to 0} \frac{- h}{h\sqrt{x}\sqrt{x + h}\left( \sqrt{x} + \sqrt{x + h} \right)}\]
\[ = \lim_{h \to 0} \frac{- 1}{\sqrt{x}\sqrt{x + h}\left( \sqrt{x} + \sqrt{x + h} \right)}\]
\[ = \frac{- 1}{\sqrt{x}\sqrt{x}\left( \sqrt{x} + \sqrt{x} \right)}\]
\[ = \frac{- 1}{x \times 2\sqrt{x}}\]
\[ = \frac{- 1}{2 x^\frac{3}{2}}\]
\[ = - \frac{1}{2} x^\frac{- 3}{2} \]
\[\]
APPEARS IN
संबंधित प्रश्न
Find the derivative of x2 – 2 at x = 10.
Find the derivative of 99x at x = 100.
Find the derivative of x5 (3 – 6x–9).
Find the derivative of `2/(x + 1) - x^2/(3x -1)`.
Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):
(ax + b)n
Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):
`(sin x + cos x)/(sin x - cos x)`
Find the derivative of f (x) = 3x at x = 2
\[\frac{2}{x}\]
\[\frac{x + 1}{x + 2}\]
\[\frac{1}{\sqrt{3 - x}}\]
\[\frac{2x + 3}{x - 2}\]
Differentiate of the following from first principle:
eax + b
Differentiate of the following from first principle:
− x
Differentiate of the following from first principle:
\[\cos\left( x - \frac{\pi}{8} \right)\]
Differentiate each of the following from first principle:
\[\sqrt{\sin 2x}\]
Differentiate each of the following from first principle:
\[\sqrt{\sin (3x + 1)}\]
Differentiate each of the following from first principle:
\[e^\sqrt{2x}\]
Differentiate each of the following from first principle:
\[3^{x^2}\]
x4 − 2 sin x + 3 cos x
3x + x3 + 33
\[\frac{( x^3 + 1)(x - 2)}{x^2}\]
\[\frac{(x + 5)(2 x^2 - 1)}{x}\]
\[\text{ If } y = \left( \sin\frac{x}{2} + \cos\frac{x}{2} \right)^2 , \text{ find } \frac{dy}{dx} at x = \frac{\pi}{6} .\]
\[\text{ If } y = \left( \frac{2 - 3 \cos x}{\sin x} \right), \text{ find } \frac{dy}{dx} at x = \frac{\pi}{4}\]
xn tan x
(x3 + x2 + 1) sin x
x5 ex + x6 log x
x4 (5 sin x − 3 cos x)
x5 (3 − 6x−9)
x−4 (3 − 4x−5)
\[\frac{e^x + \sin x}{1 + \log x}\]
\[\frac{2^x \cot x}{\sqrt{x}}\]
\[\frac{\sqrt{a} + \sqrt{x}}{\sqrt{a} - \sqrt{x}}\]
\[\frac{x^5 - \cos x}{\sin x}\]
\[\frac{x + \cos x}{\tan x}\]
\[\frac{x}{\sin^n x}\]
\[\frac{ax + b}{p x^2 + qx + r}\]
If |x| < 1 and y = 1 + x + x2 + x3 + ..., then write the value of \[\frac{dy}{dx}\]
Mark the correct alternative in of the following:
If \[f\left( x \right) = \frac{x - 4}{2\sqrt{x}}\]
Mark the correct alternative in of the following:
If \[f\left( x \right) = x^{100} + x^{99} + . . . + x + 1\] then \[f'\left( 1 \right)\] is equal to
